Alfvén eigenmodes and their destabilization by energetic ions in stellarators, mainly, in the Large Helical Device (LHD) [M. Fujiwara, K. Kawahata, N. Ohyabu et al., Nucl. Fusion 41, 1355 (2001)] plasmas, are considered. A general expression for the instability growth rate is derived, which generalizes that obtained by Ya. I. Kolesnichenko et al. [Phys. Plasmas 9, 517 (2002)] by taking into account the finite magnitude of the perturbed longitudinal magnetic field. The structures of the Alfvén continuum and Alfvén eigenmodes, as well as the resonances of the wave–particle interaction, are studied. A numerical simulation of the destabilization of Alfvén waves with low mode numbers during neutral-beam injection in a particular LHD shot is carried out. The obtained solutions represent even and odd core-localized toroidicity-induced Alfvén eigenmodes, the calculated frequencies and the mode numbers being in agreement with experimental data. The growth rates of the instabilities are calculated.